Stability for Abstract Evolution Equations
by
Patrizia Pucci
Università di Perugia
Perugia
and
James Serrin
University of Minnesota
Minneapolis
Dedicated to Carlo Pucci on the occasion of his seventieth birthday
§1. Introduction
In a recent paper [6] we studied the question of asymptotic stability for non-autonomous dissipative wave systems. Earlier work in the same direction is due to Marcati [2,3]
and Nakao [4], who treated particularly the case of abstract evolution equations. In this
note we give a new asymptotic stability theorem which extends the analysis in [2,3,4] by
taking into account the techniques introduced in [6].
We focus on abstract equations of the form
(1.1)
[P (u0 (t))]0 + A(u(t)) + Q(t, u0 (t)) + F (u(t)) = 0,
where A, F, P and Q are nonlinear operators on appropriate Banach spaces. We understand
P to be the evolution operator, A as a differential operator of divergence form, Q as a
damping term, and F as a restoring force. Concrete examples of (1.1) include the principal
case of wave systems, where P = I, A = −∆, and also, more generally, the p-Laplacian,
where A = −∆p , p > 1, as well as the polyharmonic operator A = (−∆)L , L ≥ 1.
As an important feature of the present work, we allow the damping Q = Q(t, v) to be
strongly non-autonomous in t and nonlinear in v. Our main theorem is given in Section 3,
while Section 2 is devoted to preliminary results. In particular, in Section 2 we formulate
a careful definition of solution of (1.1), which clarifies and generalizes the corresponding
definitions in [2,3,4], and moreover resolves the principal difficulty in treating the abstract
case, namely that an appropriate energy balance for (1.1) cannot be derived directly, but
must instead be inferred from analogy with concrete equations and systems.
1
§2. Notation and Preliminary Results
Let V = (V, k · kV ), W = (W, k · kW ) and X = (Xk · kX ) be real Banach spaces,
V , W 0 , X 0 their dual spaces, and h·, ·iV , h·, ·iW , h·, ·iX the natural dual pairings. Moreover,
we suppose that the spaces V, W, X have a common subspace G 6= {0}.∗
0
A : W → W 0,
F : X → X0
and P : V → V 0
be functions satisfying the following structural conditions:
(S1) A, F and P are the Fréchet derivatives of real valued C 1 potentials
A : W → R,
F : X → R,
P : V → R,
respectively, where without loss of generality we assume that A(0) = 0, F(0) = 0 and
P(0) = 0;
(S2) hA(u), uiW + hF (u), uiX ≥ 0 for all u ∈ G;
(S3) P ∗ (v) = hP (v), viV − P(v) ≥ 0 in V ;
(S4) for every d > 0 the sets
D = {u ∈ G : A(u) + F(u) ≤ d},
E = {v ∈ V : P ∗ (v) ≤ d}
are bounded in X and V , respectively, and the sets F (D), P (E) are bounded in X 0 and
V 0.
It is easily seen that
Z
A(u) =
1
Z
hA(su), uiW ds,
F(x) =
0
1
hF (sx), xiX ds;
0
consequently condition (S2) implies that
(S2)0
A(u) + F(u) ≥ 0 for u ∈ G.
Denote the time set [0, ∞) by J. Let S be a given subset of J × V and
Q : S → X0
a continuous function.
∗
If V, W, X are themselves subspaces of a common vector space Z, then we can take
G = V∩ W∩ X, this being the maximal subspace contained in all three spaces. As an
example, references [3],[4] deal with the special case W = X ⊂ V (in our notation), for
which the appropriate subspace G is simply X itself.
2
We consider the abstract evolution equation
[P (u0 (t))]0 + A(u(t)) + Q(t, u0 (t)) + F (u(t)) = 0,
(2.1)
t ∈ J.
Let K denote the subset of all functions u : J → G such that
u ∈ C 1 (J → V ) ∩ C(J → W ) ∩ L∞
loc (J → X).
We say that u is a strong solution of (2.1) if
(a) u ∈ K and (t, u0 (t)) ∈ S for a.a. t ∈ J;
(b) u verifies (2.1) in the following distribution sense:
hP (u
0
(s)), φ(s)iV |t0 =
Z
t
{hP (u0 (s)), φ0 (s)iV − hA(u(s)), φ(s)iW
0
− hQ(s, u0 (s)), φ(s)iX − hF (u(s)), φ(s)iX }ds
for all φ ∈ K and t ∈ J.
Note that the first two integrands on the right hand side of (b) are well-defined and
integrable on [0, t], t ∈ J. We show at the end of this section, with the help of further
structural conditions, that the last two terms also are meaningful.
Let E : J → R be the total energy of the field u ∈ K, that is
(2.2)
E(t) = P ∗ (u0 (t)) + A(u(t)) + F(u(t)).
(The energy function E, of course, arises naturally for classical conservation laws.) For
simplicity in printing, we shall write
H(t) = −E(t).
We now postulate the following crucial connection conditions between Q, u0 and E 0 (see
[1,6]):
(S5) For every strong solution u of (2.1) the corresponding function H is non-decreasing and
absolutely continuous on J. Moreover there are exponents q > m > 1 and a non-negative
function δ ∈ L1loc (J) - independent of the solution u - such that
(2.3)
0
kQ(t, u0 (t))kX 0 ≤ [δ(t)]1/m [H0 (t)]1/m + [δ(t)]1/q [H0 (t)]1/q
0
a.e. in J,
where m0 and q 0 respectively denote the Hölder conjugates of m and q.
(S6) There is a non-negative function σ on J, with 1/σ ∈ Lm−1
loc (J), and a function ω =
ω(τ ), τ ≥ 0, with ω(0) = 0, ω increasing in [0, 1] and ω(τ ) = 1 for τ ≥ 1, such that
σ(t)ω(ku0 (t)kV ) ≤ H0 (t)
3
a.e. in J
for every solution u of (2.1).
The regularity and monotonicity conditions expressed in (S5) are motivated by the
classical conservation balance
Z
H(t) − H(0) =
(2.4)
t
hQ(s, u0 (s)), u0 (s)ids,
0
where h·, ·i denotes an appropriate pairing between Q and v. That Q represents a damping
is expresed by the requirement that the integrand in (2.4) be non-negative; we are thus led
directly to the axiomatic assumption that H be absolutely continuous and non-decreasing
in J along any solution u of (2.1).
In concrete subcases of (2.1), when V, W and X are standard Lebesgue or Sobolev
spaces, the pairing h·, ·i reduces to an appropriate Lebesgue integration (see [6]).
An abstract case which further clarifies (2.4) occurs when S = J × Y and Y is a
Banach space with continuous inclusions X ⊂ Y ⊂ V , and Q : S → Y 0 . Here the pairing
h·, ·i can be specified explicitly as h·, ·iY . Even more, in this case (2.3) can be derived, up
to a multiplicative factor, from the direct conditions
kQ(t, v)kY 0 ≤ δ(t)(kvkm−1
+ kvkq−1
Y
Y )
and
kQ(t, v)kY 0 · kvkY ≤ γhQ(t, v), viY ,
γ = Const. ≥ 1
(reverse pairing inequality) as one easily checks. It is essentially this case which was
discussed in [3,4]; see also the remarks at the end of the paper.
We can now show that the third and fourth integrands in (b) are meaningful for all
t ∈ J. First, by (S5), Hölder’s inequality, and the fact that φ ∈ L∞
loc (J → X) we have
Z
(2.5)
0
" Z
t
|hQ(s, u0 (s)), φ(s)iX |ds ≤ sup kφ(s)kX
1/m Z
t
δ(s)ds
[0,t]
0
Z
+
t
0
1/q0 #
H (s)ds
δ(s)ds
0
1/m0
H0 (s)ds
0
1/q Z
t
t
< ∞,
0
as required. That the final integral in (b) is well-defined is a consequence of the first part
of (2.9) in the following lemma, which will also be useful for the main result of the next
section.
LEMMA. Let (S1)-(S5) hold. Then for any strong solution u of (2.1) we have
(2.6)
(2.7)
0 ≤ E(t) ≤ E(0) in J,
E 0 (t) ≤ 0 a.e. in J,
H0 ∈ L1 (J),
4
(2.8)
0 ≤ P ∗ (u0 (t)) ≤ E(0)
0 ≤ A(u(t)) + F(u(t)) ≤ E(0),
in J.
Moreover, there exists a constant C > 0 such that
(2.9)
ku0 (t)kV , kP (u0 (t))kV 0 ≤ C
ku(t)kX , kF (u(t))kX 0 ≤ C,
in J.
PROOF. From (S2)0 , (S3) and (2.2) we get E(t) ≥ 0 in J. The condition (2.6)2 follows
directly from (S5), and in turn also E(t) ≤ E(0). By (2.6) and the fact that H ∈ AC(J),
we get (2.7).
Condition (2.8) is a consequence of (2.6)1 , (S2)0 and (S3). Relation (2.9) follows
immediately from (S4), with d = E(0).
A final structure condition will be required for our main result (see [6], Lemma 3.4).
(S7) For all ` > 0 there exists ᾱ = ᾱ(`) > 0 such that u ∈ G and A(u) + F(u) ≥ ` implies
hA(u), uiW + hF (u), uiX ≥ ᾱ(`).
§3. Asymptotic Stability
We now turn to the main result of the paper.
THEOREM. Let (S1)-(S7) hold. Suppose that there is a non-negative function k 6≡ 0 of
class AC(J) such that
Z t
Z t
0
(3.1)
lim
|k (s)|ds
k(s)ds = 0,
t→∞
Z
(3.2)
lim inf
t→∞
0
0
t
(δ + σ
1−m
m
Z
)k ds
0
t
m
< ∞.
kds
0
Then along any bounded (in G) strong solution u of (2.1) we have
(3.3)
lim E(t) = 0.
t→∞
REMARK. It is worth noting that (3.1) implies k 6∈ L1 (J). [Otherwise, we would have
k 0 (t) = 0 on J and k(t) = Const. But then k must be identically zero on J, in contradiction
with the assumption k 6≡ 0.] On the other hand, (3.1) obviously holds if k 6∈ L1 (J) and
k 0 ∈ L1 (J). The last two conditions were in fact principal requirements in the main
stability Theorem 3.1 of [6], so that (3.1) gives a better condition for stability than that
result.
Various applications of condition (3.2) are given in Section 5 of [6], to which the reader
is referred.
PROOF OF THEOREM. Suppose for contradiction that (3.3) fails along some strong
solution u of (2.1). Then by (2.6) there exists ` > 0 such that E(t) & 2` as t → ∞.
5
In what follows, by a standard approximation procedure we can suppose without loss
of generality that k ∈ C 1 (J). Then since u ∈ K, we also have φ = ku ∈ K. Put
U = U (t) = hP (u0 ), φiV = khP (u0 ), uiV .
(3.4)
By (b) with φ = ku ∈ K, it follows that
t
U (s)T =
Z
t
{k 0 hP (u0 ), uiV + k[hP (u0 ), u0 iV + P ∗ (u0 )]
T
− k[P ∗ (u0 ) + hA(u), uiW + hF (u), uiX ] − khQ(s, u0 ), uiX }ds
(3.5)
Z
t
{I1 + I2 + I3 + I4 }ds,
=
T
this being valid for all t ≥ T ≥ 0 in J.
By (2.9) we have
Z
t
Z
t
I1 ds ≤
(3.6)
T
0
0
|k (s)| · kP (u (s))kV 0 · ku(s)kV ds ≤ C
2
T
Z
t
|k 0 (s)|ds.
T
Next, note that
I2 ≤ [C 2 + E(0)]k
in J,
by (2.8) and (2.9). Now fix ϑ > 0. By (S1) and (S3) the functions P and P ∗ are continuous
on V , and P ∗ (0) = 0. Hence there exists Λ(ϑ) > 0 such that
hP (v), viV + P ∗ (v) ≤ ϑ
for all v ∈ V with kvkV ≤ Λ(ϑ). Consequently

in J1
ϑ
I2 ≤ k

C + E(0) in J2 ,
where J1 = {t ∈ J : ku0 (t)kV ≤ Λ(ϑ)} and J2 = J\J1 . Now by (S6)
H0 (t)
≥ ω(Λ(ϑ))
σ(t)
for a.a. t ∈ J2 .
Thus
I2 ≤ k

ϑ
 γ(ϑ)[H0 σ]1/m0
=

 ϑk
in J1
0

γ(ϑ)[σ 1−m k m ]1/m [H0 ]1/m
where
0
γ(ϑ) = [C + E(0)] ω(Λ(ϑ))−1/m .
6
in J2 (a.e.),
Therefore, for all t ≥ T ≥ 0,
Z
t
Z
t
I2 ds ≤ ϑ
(3.7)
Z
t
k(s)ds + γ(ϑ)(T )
T
σ
T
1−m m
1/m
k ds
0
in view of Hölder’s inequality and (2.7), where
Z
(3.8)
∞
H0 (s)ds
(T ) =
1/m0
→0
as T → ∞.
T
Next, if P ∗ (u0 (t)) ≥ ` at some t ∈ J, then by (S2)
I3 (t) ≤ −`k(t).
On the other hand, if P ∗ (u0 (t)) ≤ `, then
A(u(t)) + F(u(t)) ≥ `,
since E = P ∗ (u0 ) + A(u) + F(u) ≥ 2` on J. Hence by (S7) and the fact that u(t) ∈ G, we
get
hA(u(t)), u(t)iW + hF (u(t)), u(t)iX ≥ ᾱ(`).
But P ∗ (u0 (t)) ≥ 0 from (S3), which therefore gives I3 (t) ≤ −ᾱ(`)k(t). Consequently
I3 ≤ −αk,
(3.9)
α = α(`) = min{`, ᾱ(`)}.
Let M (t) = sup{k(s) : s ∈ [T, t]}. By Young’s inequality and the fact that q > m > 1
we obtain
Z t
1/m
Z t
1/q
Z t
1/q
m
(q−m)/q
m
q
.
≤ M (t) +
δk ds
≤ M (t)
δk ds
δk ds
T
T
T
Thus, as in (2.5) with φ = u ∈ K, we have
" Z
Z
t
t
I4 ds ≤ sup ku(s)kX
[0,t]
T
1/m0
1/m Z t
0
δk ds
·
H ds
m
T
T
Z
(3.10)
t
+
1/q Z t
1/q0 #
δk q ds
·
H0 ds
T
" Z
≤ 1 (T )
t
δk m ds
T
1/m
#
+ M (t) ,
0
where
" Z
(3.11)
∞
1 (T ) = C
H0 ds
1/m0
Z
∞
+
T
T
7
H0 ds
1/q0 #
→0
as T → ∞.
Combining (3.5)-(3.7), (3.9), (3.10) yields, for all t ≥ T ≥ 0,
Z t
1/m
Z t
Z t
t
2
0
1−m m
U (s)|T ≤ C
|k (s)|ds + θ
k(s)ds + γ(ϑ)(T )
σ
k ds
T
T
0
" Z
#
(3.12)
1/m
Z
t
−α
t
δk m ds
k(s)ds + 1 (T )
T
+ M (t) .
0
In view of (3.2) there is a number B > 0 and a sequence ti % ∞ as i → ∞ such that, for
all i,
Z ti
m
Z ti
1−m m
(3.13)
(δ + σ
)k ds ≤ B
k(s)ds
.
0
0
Moreover by (3.1) we can suppose as well that
Z ti
Z ti
α
0
k(s)ds.
(3.14)
|k (s)|ds ≤
8C 2 0
0
Now choose ϑ = α/8 and T so large that
(3.15)
1 (T ) ≤ min{α/8M, C 2 },
(T ) ≤ α/8Bγ(ϑ),
which can be done by (3.8) and (3.11). Then from (3.12)-(3.15) there results, for all i such
that ti > T ,
Z
Z
Z
α ti
α ti
α ti
ti
k(s)ds +
k(s)ds +
k(s)ds
U (s) |T ≤
8 0
8 T
8 0
Z ti
Z
α ti
(3.16)
−α
k(s)ds +
k(s)ds + C 2 M (ti )
8
T
0
Z T
Z ti
7
α
k(s)ds −
k(s)ds + C 2 M (ti ).
= α
8 0
2 0
On the other hand, for ti > T ,
(3.17)
|U (ti )| ≤ k(ti ) kP (u0 (ti ))kV 0 ku(ti )kV ≤ C 2 M (ti )
and
Z
(3.18)
ti
M (ti ) ≤ k(T ) +
T
α
|k (s)|ds ≤ k(T ) +
8C 2
0
Z
ti
k(s)ds
0
by (3.14). Hence by (3.16)-(3.18) one easily obtains, for all i sufficiently large,
Z T
Z
7
α ti
2
0 ≤ U (T ) + 2C k(T ) + α
k(s)ds −
k(s)ds.
8 0
4 0
Consequently, since k is non-negative, we find that k ∈ L1 (J).
This provides an immediate contradiction since (3.1) implies that k 6∈ L1 (J), and so
completes the proof.
8
§4. Concluding Remarks
1. The special case S = J × Y discussed in Section 2 was treated earlier by Nakao
[4]. For his main result it was assumed that V is a Hilbert space, that W = X, and that
P = I, A + F = FA : X → R+
0 . In addition, (S5) and (S6) were formulated with the
restrictions
0 < Const. ≤ δ(t) = Const. σ(t) in J,
ω(τ ) = τ m ,
m ≥ 2;
and (S7) for the explicit choice ᾱ(`) = Const. `. Finally, Nakao considered exactly the
special function k ≡ 1, for which (3.1) and (3.2) are automatically satisfied.
The case when S = J × Y was also discussed by Marcati [3], but with autonomous
damping. The hypotheses in [3] are similar to those in [4], with the exceptions that P need
not be the identity, nor V a Hilbert space.
2. If W is continuously imbedded in X, slightly generalizing the assumption W = X
in [3,4], it is easy to see that the solution space K reduces to C(J → W ) ∩ C 1 (J → V ).
In this case, from the relation F ∈ C(X → X 0 ) we get
F ◦ u ∈ C(J → X 0 )
for all u ∈ K,
which is enough to make the final integral in (b) well-defined. Hence the assumption in
(S4) that F (D) be bounded in X 0 can be omitted, since its only purpose was to make the
relation (b) meaningful. (Of course, when (S4) is thus modified, the consequent implication
in (2.9), that kF (u(t))kX 0 ≤ C along a strong solution of (2.1), must also be dropped.)
3. Condition (S6) need not hold in the entire interval J, but in fact can be restricted
to a measurable control subset I ⊂ J. The main theorem then needs to be revised only in
two places. First, (3.1) should be modified to include the condition
(3.1)0
k=0
in J \ I,
and second, (3.2) should be replaced by
0
(3.2)
Z
(δ + σ
lim inf
t→∞
1−m
m
Z
)k ds
[0,t]∩I
t
m
kds
< ∞.
0
The proofs remain almost the same.
The importance of control subsets in studying asymptotic stability is illustrated in
[5], in the context of ordinary differential systems.
Acknowledgment. P. Pucci is a member of Gruppo Nazionale di Analisi Funzionale e
sue Applicazioni of the Consiglio Nazionale delle Ricerche. This research has been partly
supported by the Italian Ministero dell’Università e della Ricerca Scientifica e Tecnologica.
9
REFERENCES
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Equations with Dissipation, to appear.
[2] P. Marcati, Decay and stability for nonlinear hyperbolic equations, J. Diff. Equations
55 (1984), 30-58.
[3] P. Marcati, Stability for second order abstract evolution equations, Nonlinear Anal. 8
(1984), 237-252.
[4] M. Nakao, Asymptotic stability for some nonlinear evolution equations of second order
with unbounded dissipative terms, J. Diff. Equations 30 (1978), 54-63.
[5] P. Pucci & J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM J. Math. Anal. 25 (1994), 815-834.
[6] P. Pucci & J. Serrin, Asymptotic stability for non-autonomous wave systems, to appear.
10